A Framework For Community Identification in Dynamic Social Networks

1
A Framework For Community Identification in
Dynamic Social Networks

• Chayant Tantipathananandh
• Tanya Berger-WolfDavid KempePresented by
  Victor Lee

2
Outline of Presentation

• The Challenge Dynamic Social Networks
• Framework and Problem Formulation
• Individual and Group Colorings
• Group Coloring Heuristics
• Experimental Results
• Future Directions

3
The Problem

• Many well-known approaches to identify
  communities in social networks
• Graph Partitioning
• Clustering
• Various measures of closeness or density
• But, these approaches generally assume static
  networks
• Most social networks are dynamic

4
Dynamic Social Networks

• Social Networks change over time
• Membership changes
• Interaction changes
• Most community identification techniques
• Use a single snapshot
• Or use time-averaged measurements
• Lose important information

5
Importance of Dynamic Information
T1 T2 T3 T4 T5 T6
A
B
A
B
A
B
C
A
B
A
B
A
B
time
A
B
C
A
B
C
A
B
A
B
C
A
B
C
A
B
C
Network 1 Network 2

• Networks 1 and 2 same average characteristics,bu
  t
• Network 1 shows an oscillation
• Network 2 suggests that C joins the community

6
Proposal

• New framework for modeling social networks over
  time
• Algorithms and Heuristics to identify dynamic
  communities
• Experiments to verify the concept and the
  computational performance

7
Problem Formation

• Given
• A set of individuals
• A sequence of snapshot observations
• Find
• A best-fit set of time-varying communities C(t)
• Best-fit time-varying community membership for
  each individual
• Approach
• Combinatorial optimization
• Graph coloring

8
Model Individuals and Groups

• Set of individuals X i1, i2, in
• Sequence of observations ltP1, P2, PTgt
• Discrete time
• Record interaction between individuals
• The set of individuals interacting at time t
  define a group.
• If A interacts with B, and B interacts with
  C,than A,B,C ? a group

A
C
B
9
Group vs Community

• Snapshot Graph
• Individual is a vertex
• Interaction is an edge
• Group is a connected subgraph
• Assumption interaction is sufficiently limited
  so that the graph is not connected (we have
  disjoint groups)
• Group ? Community
• Groups capture observed interaction at a point in
  time
• Communities extend over time

10
Graphing the Observations

• Each time slice is one observation
• Edges within a time slice show observed
  interaction at time t
• Add edges joining all observations of the same
  individual
• No edges between groups from one time to another

? individual ? group
11
Refine the Problem

• A community appears as a sequence of groups, of
  at most one group per time slice.
• Tasks
• Assign each group to a community(color the group
  vertices)
• Assign each individual to a community, for each
  time step (color individual vertices)
• More Assumptions
• Individuals belong to one community at a time
• Individuals dont change community frequently
• Individuals frequently appear in their community

12
Cost Model

• Quantify a good community identification
• Assign costs to undesirable behavior
• I-cost ? when an individual changes color.
• G-costs
• b1 when an individual is absent from its
  community.
• b2 when an individual is present in a different
  community.
• C-cost g for each color that I uses
• Find a coloring with minimum cost

13
Coloring Choices and Costs
At time T3, C temporarily changes its interaction.
T1 T2 T3 T4
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
time
A
B
C
D
A
B
C
D
Coloring 1 Coloring 2

• Coloring 1 C changes community and then changes
  back.
• Cost 2a ( g if this color hasnt been used
  before)
• Coloring 2 C stays in its original community and
  just visits.
• Cost b1 b2
• Optimal coloring depends on comparison (b1
  b2) lt (2a g) or (2a)

14
Finding Optimal Colorings

• Finding the optimal solution is NP-hard
• Partition the problem
• Find an optimal set of communities
• Find optimal assignment of individuals to
  communities
• If Phase 1 (Group Coloring) is completed first
• Phase 2 is reduced from O(2N) to O(2G),N of
  individuals, G of groups
• The cost incurred by one individuals coloring
  is independent of the colors chosen by others.

15
Independence of Individual Color Choice

• Proof
• Cost of an individuals behavior A (I-cost)
  B (G-cost) C (C-cost)
• Costs are assessed individually
• I-cost a ( of color changes)
• G-cost b1 ( absences from its group) b2
  ( visits to other groups)
• C-cost g ( of colors that an individual
  uses)
• So, we can solve for each individual one at a
  time.
• Moreover, we can assess cost incrementally,from
  time t to time t1

16
Individual Coloring Algorithm

• C set of all colors observed to be used by an
  individual i
• F(t) S ? C 1 S t all possible subsets
  of colors up to time t
• G(t,x) G-cost to use color x at time t
• I(t,x,y) I-cost to use color x at time t-1 and
  color y at time t
• C(x,R) C-cost to use color x when color set R
  has been used
• Min. cost at time t, using color x, with color
  set S used
• At time1 G(I, x, x) G(1,x) At timet G
  (t, S, x) G(t, x) min G(t-1, R, y)
  I(t, x, y) C(x, R) over all R and y,
  where R ? F(t-1), y ? R R U x S,

i-cost changing colorg-cost wrong
groupc-cost new color
17
Optimal Individual Coloring

• Given a group coloring, the minimum cost of
  coloring the individual I is min G(T, S, x) S
  ? F(T), x ? S
• Time complexity is O( nTC2 2C )
• Space requirement is O( C 2C )
• If the number of groups C is not large, the
  complexity is tractable.

18
Optimal Group Coloring

• Determine the best mapping of groups at time t to
  groups at time t1
• Groups that are mapped across time are part of
  the same community and have the same color
• A coloring is good if most individuals can retain
  their color from step to step.

19
Bipartite Matching Heuristic

• Matching Graph
• For each pair of groups g, g at times t, tt1,
  add a weighted edge from vg,t to vg,t
• Weight g n g (similarity of g to g)
• Find the maximum weight bipartite matching
• Evaluation
• Weights i-cost more than g-cost
• Performs well if membership is fairly stable
• No long range perspective
• More efficient heuristics?

i-cost changing colorg-cost wrong
groupc-cost new color
20
Greedy Heuristics for Group Coloring

• Approach Maximize pairwise similarity between
  groups, for all pairs of groups over all
  timesteps
• Jaccards index Jac(g, g') g n g' g U
  g'
• Weighted for temporal proximity JacD(g, g')
  Jac(g, g') t - t'

overlap between g and g', scaled to size of g and
g'
21
Greedy Heuristics for Group Coloring

• Greedy Heuristic 1 (time is not a factor)
• Construct a square similarity matrix of size
  groups
• Using agglomerative clustering
• Greedy Heuristic 2 (look backwards in time) For
  t1 to T do
• Match most similar pairs g, g' for any time t' lt
  t
• If similarity0 or all colors have been used, add
  a new color
• Greedy Heuristic 3 (look back the shortest
  interval)
• Like Heuristic 2, but use t', t' is the closest
  value to t such that ? similarity(g, g') gt 0

22
Experiment 1 Verify the Framework

• Does the framework capture the intuitive concept
  of dynamic community?
• Procedure
• Construct small, synthetic datasets
• Use exhaustive search to get a truly optimal
  coloring

23
Experiment 1A Assembly Line

• At each time step, 1 member leaves and 1 enters a
  group, resulting in a complete membership change
  in 3 steps.

• Results change as costs change. (A) favors
  stable membership. (B) allows for more fluid
  membership.

24
Experiment 1B Dutiful Children

• 2, 3, and 4 are Children. 0 and 1 are Parents
  that visit a different child each timestep.

• Results Framework succeeds at detecting the
  individual children as well as the visitation
  pattern.

25
Experiment 2 Quality of Heuristic Results

• Do the heuristics obtain colorings similar to
  those of an exhaustive search?
• Procedure
• Re-test the synthetic datasets using the various
  heuristics

Results At least one Heuristic method obtains
the same coloring and total cost as Exhaustive
Search
26
Experiment 3 Real World Datasets

• Do the framework and heuristics together obtain
  expected results using real-world datasets?

27
Experiment 3A Southern Women

• Eighteen women in 1933 in Natchez, Tennessee
• Tracks their attendance at 14 social events

28
Experiment 3A Prior Results

• Twenty one analyses (1941 to 2001) all show
  similar results
• Two clear communities
• The membership of individuals 8, 9, and 16 is
  less certain.

29
Experiment 3A Results

• Detects 4 communities, which are subsets of the
  traditional 2 communities

• Individuals 6 and 10 change membership over time
• By adjusting cost factors, the results of most of
  the 21 prior analyses can be duplicated

30
Experiment 3B Grevys Zebra

• 28-member zebra herd observed 44 times over 3
  months in 2002

• The graph to the left shows the aggregate
  interaction.
• Temporal information is lost.

31
Experiment 3B Results

• Inferred communities agree with manual results
  obtained by biologists.
• 4 stable communities
• Some short-lived communities and some visiting

32
Conclusions

• We present a framework for identifying
  communities in dynamic social networks
• The framework produces meaningful results
  compared to traditional methods
• Heuristic methods produce near-optimal solutions
• Future Directions
• Develop an approximation algorithm which
  guarantees the quality of the result
• Investigate scalability over network size and
  time
• Relax assumptions about interaction and dynamics